Journal of the Australian Mathematical Society - Series A Vol. 64 Part 2 (1998) Multiplicative isogeny estimates David Masser Mathematisches Institut Universität Basel Rheinsprung 21 4051 Basel Switzerland Abstract: The theory of isogeny estimates for Abelian varieties provides `additive bounds' of the form `d is at most B' for the degrees d of certain isogenies. We investigate whether these can be improved to `multiplicative bounds' of the form `d divides B'. We find that in general the answer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptional primes $\ell$ in connexion with modular Galois representations coming from elliptic curves: we prove that the additive bounds for $\ell$ of Masser and Wüstholz (1993) can be improved to multiplicative bounds (Theorem 3). PDF file size: 97K © Copyright 1998, Australian Mathematical Society